Ask whether list seats compensate for district seats
Two ballots or two tiers do not by themselves make a system proportional.
By the end: Distinguish parallel allocation, compensatory allocation, overhang, and leveling rules.
Independent tiers preserve district bonuses
A parallel system allocates district seats and list seats separately, then adds them. If one party receives a district-seat bonus, a proportional list tier does not necessarily cancel it. The share of seats in each tier and the list formula matter.
The lab has five district seats and five list seats. Party votes are 40,31,29; unmodified Sainte-Laguë gives 2,2,1 of the five independent list seats. Adding district wins 4,1,0 yields totals 6,3,1. Calling the list component proportional does not make the combined ten seats proportional.
Identify what each tier allocates
A mixed election composes a legislature rather than choosing one officeholder. District contests select particular representatives, while a list calculation allocates seats among parties. Those operations can share an election day without sharing an allocation target. To identify the mechanism, ask whether the list calculation knows the district results. If it allocates a separate block of seats without subtracting district wins, it cannot systematically correct those wins. A second ballot supplies information, but the formula determines how that information affects the legislature.
Keep two input records separate in this example. The party-vote record is A 40, B 31, C 29, out of 100 valid party votes. The district-result record is A four seats, B one, C none, out of five district seats. The first record measures votes; the second already measures elected representatives. There are no district vote totals here. You cannot infer that A received 80% of district ballots merely because it won 80% of district seats. Margins and turnout could differ across districts.
Now allocate the five independent list seats using unmodified Sainte-Laguë. At the start the quotients are 40, 31, and 29, so A receives the first seat. A's next quotient becomes 40 divided by three. B and then C receive seats before that quotient wins A a second seat. The fifth award compares A's 40/5, B's 31/3, and C's 29/3. B wins because 31/3 exceeds both alternatives. Thus the independent list seats are 2, 2, and 1, not the whole-assembly entitlement.
Five independent list awards
Award
A quotient
B quotient
C quotient
Winner
1
40/1
31/1
29/1
A
2
40/3
31/1
29/1
B
3
40/3
31/3
29/1
C
4
40/3
31/3
29/3
A
5
40/5
31/3
29/3
B
Quotients are party votes divided by the next odd divisor. Only the winning party changes its divisor. Awards give A 2, B 2, C 1. Exact comparisons, not rounded displays, determine winners; an exact tie would favor the earlier party A, then B, then C.
Add those seats to the district results party by party. A receives four plus two, B one plus two, and C zero plus one. The totals are 6, 3, and 1. Dividing each by ten produces seat shares of 60%, 30%, and 10%. Against party-vote shares of 40%, 31%, and 29%, the gaps are plus twenty, minus one, and minus nineteen percentage points. The list tier follows a proportional formula within its own five-seat block, yet the combined result preserves much of the district imbalance.
This distinction also explains why a vote count cannot simply be added to a seat count. Adding A's forty party votes to its four district seats produces no meaningful representation measure. Adding two independently awarded list seats to four district seats does. Likewise, combining candidate and party ballot totals into one national percentage would require a specified rule and a reason for that denominator. Where voters may support a local candidate and a different party list, the two reports need not express the same choice.
Read the comparison below as a controlled calculation, not as evidence of how voters would respond to either design. The party totals and district winners remain stipulated. Separate ballot choices, nominations, or campaigns might change them in another institutional setting. The selected mixed-system passages in International IDEA's 2005 handbook explain the independent-versus-compensating distinction. Their country examples do not establish today's law. Here the invented parties let you inspect the allocation relationship without assuming any jurisdiction's eligibility rules, candidate selection procedures, or political behavior.
Add representatives, not votes to seats
One rectangle is one seat. Independent list awards do not subtract district wins. Combining the rows gives A 6, B 3, C 1 in a ten-seat assembly; party votes remain 40/31/29 out of 100.
Compensation targets the combined result
In a simplified mixed-member proportional model, first calculate each party’s entitlement across the whole ten-seat assembly from party votes. Here the target is 4,3,3. Subtract district seats already won to find top-ups: 0,2,3. The five list seats then correct the district imbalance.
This distinguishes the mechanical role of the tiers. Actual systems may use separate ballots, eligibility thresholds, regional entitlements, and different party/candidate rules. A district candidate’s victory and the party-vote allocation are related by a legal specification, not by the mere presence of two marks on a ballot.
Calculate compensation against a common target
Compensation starts with a different question. Instead of asking how to distribute five additional seats independently, ask how many of all ten seats each party should receive. Apply Sainte-Laguë to the same party votes with ten awards. This gives a target of 4, 3, and 3. District representatives then count toward those entitlements. Under this specification they are not an extra reward added after a full proportional allocation. The target constrains the combined party totals whenever the district wins do not exceed it.
You can verify the target without trusting its final row. After five awards the allocation is 2, 2, and 1. C receives award six because 29/3 exceeds A's 40/5 and B's 31/5. A receives award seven at 40/5. B receives award eight at 31/5. C receives award nine at 29/5, which exceeds A's 40/7 and B's 31/7. A receives award ten at 40/7. These comparisons give 4, 3, and 3, with every quotient expressed as votes divided by a stated divisor.
Subtract district seats only after establishing that target. For A, four minus four leaves no top-up. For B, three minus one leaves two. For C, three minus zero leaves three. The top-ups sum to five, so the initial five district representatives plus these five list representatives fill ten seats. Adding district and list columns recovers 4, 3, and 3. The district result still identifies which people enter through local contests, but it no longer gives A six of the ten seats in this fixed example.
Extend the list calculation to a ten-seat target
Award
A quotient
B quotient
C quotient
Seats after award A/B/C
Starting after 5
40/5
31/5
29/3
2 / 2 / 1
6: C
40/5
31/5
29/3
2 / 2 / 2
7: A
40/5
31/5
29/5
3 / 2 / 2
8: B
40/7
31/5
29/5
3 / 3 / 2
9: C
40/7
31/7
29/5
3 / 3 / 3
10: A
40/7
31/7
29/7
4 / 3 / 3
Each award row shows the competing quotients before that award and the seat counts afterward. With district wins 4/1/0, subtracting them from target 4/3/3 gives top-ups 0/2/3. The starting row is not an additional award.
Exact proportionality would require fractional entitlements of 4, 3.1, and 2.9 seats. A legislature cannot seat a tenth of a representative under this model. The integer target therefore gives B 30% rather than 31%, and C 30% rather than 29%. Calling the outcome compensatory describes the relationship between tiers, not a promise to reproduce every percentage exactly. Nor does the target establish candidate choice within lists. An ordered list and a voter-influenced list can distribute the same party entitlements among different people.
The totals also change the arithmetic of possible legislative majorities. Assume, only for this comparison, that a decision needs strictly more than half of all filled seats and that each party votes together. In a ten-seat assembly the requirement is six. Under parallel allocation A's six seats meet it alone. Under compensation A's four do not, but B and C together have six. These are feasible seat combinations under a stated voting rule. They do not establish that B and C agree on policy, will negotiate successfully, or will form a government.
Before the lab, try explaining which subtraction the word compensation requires. If someone proposes list additions of 2, 2, and 1, ask which target those additions correct toward. Their independence is visible because the list calculation did not use district wins at all. If someone proposes 0, 2, and 3, check both the party totals and the assembly total. This method catches a common error, applying compensation to only the list tier's size rather than the whole assembly. The same word can otherwise conceal incompatible calculations.
Seat combinations that meet a six-seat requirement
One rectangle is one seat. The boundary after five seats marks half of ten, not a sufficient majority. Assuming party unity and a requirement of more than half of all filled seats, parallel A alone has six; compensatory B plus C also have six. Feasible arithmetic is not a coalition forecast.
Work through the mechanism
Independent versus compensating list seats
Party votes remain 40 / 31 / 29. Whole-assembly target at ten seats: 4 / 3 / 3. Changing the scenario replaces district wins in both panels.
Both bar scales run from 0% to 100% of each panel’s stated assembly size. Counts are exact; displayed percentages are rounded. The second model keeps district wins, tops other parties up to their original targets, and retains overhang without leveling. It is not a specification of a current country’s complete mixed system.
Rule
Districts A/B/C
List additions A/B/C
Final seats
Parallel
4 / 1 / 0
2 / 2 / 1
6 / 3 / 1
Compensatory model
4 / 1 / 0
0 / 2 / 3
4 / 3 / 3
Retain overhang, no leveling
5 / 0 / 0
0 / 3 / 3
5 / 3 / 3, total 11
The overhang row states a hypothetical policy, not a current jurisdiction’s complete law.
Use the two district scenarios to isolate the rule
Begin the lab with the district preset A 4, B 1, C 0. Read across each party's district, independent-list, and compensatory-top-up entries. Predict the two final totals before looking at the bars. A's district four plus independent-list two gives six on the parallel side. Its target four minus district four gives zero list additions on the compensatory side. Both panels contain ten seats in this setting, so a two-seat difference also means a twenty-percentage-point difference. Different assembly sizes break that equivalence.
Next choose the preset in which A wins all five districts. Party votes stay 40, 31, and 29. The independent list allocation therefore stays 2, 2, and 1. Parallel totals become 7, 2, and 1, still summing to ten. The ten-seat compensatory target also stays 4, 3, and 3, because its party-vote input has not changed. Only district wins changed. This is a clean comparison of allocation mechanics with fixed party votes, not a reconstruction of how a five-district sweep would arise from actual district ballots.
Ordinary subtraction now exposes a conflict. A's target four minus district five equals negative one. A negative list seat cannot remove a district representative while preserving the model's promise to keep every district winner. The stated policy replaces that negative top-up with zero. B and C each receive three list seats, producing final totals 5, 3, and 3. These totals require six list additions rather than the initial five. The assembly expands to eleven because the model retains A's overhang and does not reduce the other parties' original targets.
A negative difference exposes the policy choice
Party
Ten-seat target
District wins
Target minus districts
Actual top-up
Final seats
A
4
5
−1
0
5
B
3
0
3
3
3
C
3
0
3
3
3
Total
10
5
5 net
6 seats added
11
The rule keeps every district winner and replaces negative additions with zero. The signed differences sum to five, but the executable nonnegative additions sum to six. This is retained overhang without leveling.
Calculate shares again using eleven as the denominator. A has 5/11, about 45.45%; B and C each have 3/11, about 27.27%. Neither three-seat party now holds 30% of the assembly. A's excess relative to its original target is one seat, while its seat-share excess over 40% of party votes is about 5.45 percentage points. Those are different descriptions with different units. Keeping eleven as the denominator explains why a panel can show five seats without meaning half the assembly, even though five was half the initial ten-seat size.
Compare the two bars using their common percentage axis rather than imagining that identical seat counts have identical widths. In the sweep setting, parallel A has seven of ten seats, or 70%, whereas compensated A has five of eleven, about 45.45%. B has two of ten versus three of eleven. The exact counts and panel totals carry information that rounded percentages alone would lose. There is no control for leveling seats in this lab. Reading the retained-overhang result as though leveling had occurred would silently change the model.
Finally return to the first preset and identify what reverses. District wins revert, independent list awards stay fixed, and the nonnegative top-ups again sum to five. The assembly returns to ten seats. If you can explain those changes without referring to bar colors, you have identified the rule. A useful written prediction records inputs, target, additions, and final denominator separately. It should not claim that a party gained voters when it gained seats here. The scenario changes seat inputs directly and supplies no new voter-level evidence.
The sweep uses two different assembly totals
Party
Party votes out of 100
Parallel seats out of 10
Retained-overhang seats out of 11
A
40 = 40%
7 = 70%
5 ≈ 45.45%
B
31 = 31%
2 = 20%
3 ≈ 27.27%
C
29 = 29%
1 = 10%
3 ≈ 27.27%
Exact seat fractions determine the shares. Rounded percentages in the eleven-seat column sum to 99.99%, not because a seat is missing. Party votes and seats have different denominators.
Handle overhang explicitly
If A wins all five districts but its proportional entitlement is four, subtracting produces a negative top-up for A. The model cannot remove a won district seat under its assumptions. It retains A’s five, gives B and C their target three each, and expands to eleven seats. That is a retained-overhang example without leveling.
Other systems may add further leveling seats, constrain compensation, or use different legal mechanisms. An eleven-seat result of 5,3,3 is not exactly the original ten-seat target in percentage terms. Do not label it fully proportional merely because some top-ups were calculated.
Separate an overhang policy from its consequences
Retaining overhang and adding leveling seats are different decisions. This model keeps A's fifth district seat while preserving B's and C's original targets. It stops at 5, 3, and 3. A leveling procedure would need another specification that recalculates or extends entitlements for the enlarged assembly. The phrase mixed-member proportional determines neither that procedure nor its limits. You cannot derive a legally valid assembly size by appending an unspecified number of corrective seats to the eleven already counted here.
To see why the conflict requires a policy, compare an explicitly different hypothetical constraint. Suppose all district winners must remain but the assembly must contain exactly ten seats. With A already holding five, only five seats remain for B and C together. Their original targets total six. It is impossible to preserve all three demands at once, fixed size ten, A's five district wins, and both three-seat targets. This is an arithmetic incompatibility in this input, not a theorem that every mixed design must expand or that any particular legal solution is required.
Three constraints cannot all hold in the sweep
Demand
Seat requirement
Conflict with fixed ten seats
Keep all district wins
A retains 5
Leaves only 5 for B and C
Keep B and C targets
B 3 plus C 3 = 6
Needs 6 rather than the remaining 5
Keep both demands
5 + 3 + 3 = 11
Exceeds fixed size by 1
This is a constraint check, not an alternative implemented policy. The lab relaxes fixed size and stops at eleven. A different fixed-size policy would have to change another demand; no leveling calculation is performed.
There are further specifications outside this example. An eligibility threshold could change which party votes enter the target calculation. Regional compensation could require separate targets rather than one national target. Independent district winners could require distinct treatment. None of those is represented by a hidden option in the lab. To analyze a real proposal, obtain the actual sequence of eligibility tests and allocations. In particular, a threshold's percentage needs its own vote denominator; it cannot be inferred from the number of district winners or from an assembly seat share.
The two tiers also leave questions about individual representation unanswered. Four seats for a party do not tell you whether voters could alter its candidate order, whether a losing district candidate may enter on a list, or which representative handles a constituent's request. These questions require candidate and institutional rules beyond the party totals. The presence of geographically elected members demonstrates a district connection in the selection procedure. It does not measure responsiveness, access, or how much influence a representative exercises once the legislature begins its work.
Keep the evidential claims separate when comparing designs. The calculation establishes that the stipulated sweep yields parallel totals 7, 2, and 1 and retained-overhang totals 5, 3, and 3. A claim that compensation changes incentives for district nominations is behavioral and needs assumptions about parties' objectives and expectations. A claim that it improves government stability is empirical and causal, requiring a defined outcome and a credible comparison. A judgment that one result is fairer requires criteria. None follows merely because the subtraction and division are correct.
Audit a new mixed-system description by asking for both the starting assembly size and the actual final size. Reproduce the independent list awards or the whole-assembly target, then subtract district wins where the rule requires it. Check every party total and the sum of additions. If a target is exceeded, locate the explicit overhang policy instead of treating a negative top-up as an executable instruction. End with the strongest conclusion the evidence supports, a seat allocation under stated assumptions. Do not turn that result into a prediction or endorsement.
Which evidence would answer the claim?
Claim
Object to inspect
What the seat ledger cannot establish
The list tier compensates
Whole-assembly target minus district wins
Candidate responsiveness
District nominations will change
Party incentives, beliefs, and behavior
Future choices from fixed ballots
Government will be more stable
Defined stability outcome and causal comparison
A forecast from addition alone
The design is preferable
Explicit representational priorities
A unique normative ranking
Mechanical, behavioral, empirical or causal, and normative claims require different support. The invented allocation establishes only the first kind directly; current legal compliance requires the actual legal specification.
Exercises
Two tiers, different totals
Why does adding proportional list seats 2,2,1 to district wins 4,1,0 not yield the compensatory target 4,3,3?
Hint
The independent list allocation ignores the district seats already won.
Worked answer
Calculation
Result
District + independent list
6 / 3 / 1
Target − districts, then add back districts
4 / 3 / 3
The connection between tiers determines the correction.
The parallel sum is 6,3,1. Compensation instead starts with the whole-assembly target and subtracts district seats, giving top-ups 0,2,3. The two formulas answer different questions: a fair share of an independent list tier versus a correction toward combined entitlements.
Count the overhang assembly
A wins five districts against a target of four. Under the stated retained-overhang rule, B and C each receive three seats. What is the total and A’s seat share?
Hint
The denominator is no longer ten.
Worked answer
Quantity
Value
Assembly size
11
A seat share
45.45%
A party-vote share
40%
Changing assembly size changes every seat-share denominator.
The assembly has 5 + 3 + 3 = 11 seats. A has 5/11 ≈ 45.45%, compared with 40% of party votes. The retained extra seat leaves disproportionality. Additional leveling would be a separate policy requiring another calculation.
Sources and further reading
The explanations, counts, and worked answers are complete without these optional links. Reference annotations identify the selected material reviewed. Historical country examples are not statements of current law, and the bounded models are not certified election-counting software. Mechanical results, behavioral evidence, and judgments about desirable representation remain separate.
Selected explanations of system families, district magnitude, thresholds, proportional formulas, mixed systems, and design objectives. Country examples and legal thresholds are historical, not statements of current law.